On the role of discrete and continuous modes in a cooled high-speed boundary layer flow

Abstract

The disturbance flow field for a Mach$6$flat-plate boundary layer flow with a wall-to-free-stream temperature ratio of$0.5$is studied using direct numerical simulation (DNS), linear stability theory (LST) and biorthogonal decomposition. In the second-mode instability region, the DNS flow field can be reconstructed using a single LST mode, namely$F^+_1$. However, when the supersonic mode emerges, none of the discrete modes nor the continuous branches alone can precisely match the DNS data. A superposition of the pair of discrete supersonic modes$F^\pm _1$and the slow acoustic continuous spectrum is required to reproduce the disturbance amplitude distributions and the amplification rates observed in the DNS. Another finding is that the supersonic mode$F^-_1$, which should be decaying following LST, in actuality is amplified based on the projected DNS data.